Subject: Re: Open letter to those who believe in a right to free software
Date: Wed, 3 Nov 1999 11:58:43 -0500

Bernard Lang wrote:
[about my report on how math works]
> Nice field observation report ...
> Now, what do you deduce from it ?
>   (assuming accuracy and generality)

You mean what lessons should we draw from it?  The biggest
ones have to do with the importance of keeping lines of
communication open between different groups if progress is
going to happen.  The importance of this improves if your
endeavor (like with free software) is using a distributed
development model.

Don't just work out how you will do things.  Keep track of how
other people are doing then and know how to explain your
stuff to them.  To name but one example I think that the EROS
essays on capabilities are invaluable - I would go so far as
to say that they currently have more impact than EROS does!
(Visit for more.)

>  - that venture capital should come in and get the field better
>    organized ?


>  - ... and be rewarded by exclusivity on using results ?


>  - that other fields do not do that (how do you know ?)
>    and are hence more efficient ?
>  - ... or is it common in sciences ? ... and in other areas of knowledge

My wife's field reports on biology, as well as experiences in
other sciences indicate that this situation is specific to math.  My
wife can pick up most biology papers, certainly anything in
molecular and cell biology, and read it.  Furthermore in many
places, such as the former Eastern Block, this is not a problem.
In all cases a clear commitment to keep lines of communication
open is key...

>    (read Sokal)   but shows more in math ?
>  - that math is indeed very difficult ?

Anything can be difficult if you make it so.

Maintaining lines of communication on technical subjects
requires work both on the part of would-be listeners and on the
part of the speaker.  This effort is worthwhile and failure by
either side to recognize this fact is dangerous to the long-term
health of your endeavor.

>  - that it is surprising there there is nevertheless progress in the
>    field (or no longer) ?

It is surprising.  There is a lot of activity, but activity is not the
same as progress.  Plus nobody knows how much is being

The question of math being lost is not trivial.  For instance I
once figured out a nice way to generate answers to such
questions as, "Which polynomial with integer coefficients
has square-root(2) plus cube-root(3) as a root?"  This was
new to everyone I showed it to except one very old number
theorist who remembered that some really old texts had
something about this.  Well it turns out that it was the original
proof and calculating such examples was a standard
exercise over a century ago.  However favored proofs
have changed and today virtually nobody knows this basic

This holds both in the study of symmetric polynomials and
algebraic number theory, both originally motivated by the
solution to my problem, both having forgotten the solution
to my problem, and both having forgotten that they ever
had any connection to each other!  (The solution is to
write in factored form a polynomial which is symmetric in
the roots of two other polynomials with integer coefficients.
Multiply it out, and after algebraic manipulation you get a
new polynomail whose coefficients are polynomials in the
coefficients of your original two.  The two you start with are
"x^2 - 2" and "x^3 - 3".  The result you will get, assuming
no careless errors, is a 6'th degree polynomial with the
desired root.)

>  - ... and it might be woth investigating how and why ?
>  - that most mathematicians are not really contributing,
>    only some of the best ones ?

This is unfortunately true.  The lesson to be drawn is the
importance of keeping lines of communication open
and not getting isolated.  It isn't that this is an inevitable
result.  It is quite preventable.  But it takes effort.

>  - ... and that you know how to identify the good ones (or you don't) ?

You don't.  I have heard many horror stories of very good
mathematicians who were polyglots who could not get
tenure for the simple reason that while they had a lot of
good papers they didn't have enough in any one area to
get recommendations.  And, of course, nobody on the
tenure committee was going to read any of the papers!

>  - ... and that the other are completely useless ?
>    or possibly they fulfill another role ?
They fill another role.

They teach calculus.

This is the main thing that math departments contribute
to our society and they are not doing a very good job
at it!  However I cannot go into that without opening up
more cans of worms.  Suffice it to say that this is a
subject of some controversy in mathematics...

Anyways I don't think that mathematics is a very good
model for free software to follow.  Imagine a community
of programmers who are unable to actually try running
or compiling a program, who have gone down a social
path which reduces the amount of peer review that
goes on.  This is furthermore a group which (for various
reasons) is popularly regarded as dealing with the
epitome of pure Truth.  Yet I can point at major results
that are widely distrusted within mathematics!  I am not
just talking about mistakes in random papers.  I am
talking about thinks like the classicification of finite
groups (an effort that took decades and is being
redone because the people who did it can't read and
don't believe their own proof) and the proof of the four
color theorem (the debate about which literally comes
down to deciding whether the bugs in the program they
wrote for their analysis came up when they ran it).

I hold free software development to higher standards
than that, and I hope that others do as well!